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Unformatted text preview: for Laplace’s equation in a disk: Find u(r, θ), 0 < r ≤ 1, such that 1∂ r ∂r r ∂u ∂r u(r, θ + 2π ) = u(r, θ), + 1 ∂2u = 0, r2 ∂θ2 u well-behaved near r = 0 and u(1, θ) = h(θ), h(θ) = 1 + cos θ − 2 sin θ + 4 cos 2θ. where 5.6.2. Solve the following boundary value problem for Laplace’s equation in a disk: Find u(r, θ), 0 < r ≤ 1, such that 1∂ r ∂r r ∂u ∂r u(r, θ + 2π ) = u(r, θ), + 1 ∂2u = 0, r2 ∂θ2 u well-behaved near r = 0 and u(1, θ) = h(θ), where h(θ) is the periodic function such that h(θ) = |θ|, for −π ≤ θ ≤ π . 5.6.3. Solve the following boundary value problem for Laplace’s equation in an annular region: Find u(r, θ), 1 ≤ r ≤ 2, such that 1∂ r ∂r u(r, θ + 2π ) = u(r, θ), r ∂u ∂r + 1 ∂2u = 0, r2 ∂θ2 u(1, θ) = 1 + 3 cos θ − sin θ + cos 2θ and u(2, θ) = 2 cos θ + 4 cos 2θ. 140 5.7 Eigenvalues of the Laplace operator We would now like to consider the heat equation for a room whose shape is given by a well-behaved but otherwise arbitrary...
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